The spring that swings when it was only bounced

The swing that is pumped, not pushed found that a child on a swing can raise the amplitude without anyone pushing, by standing and squatting twice in each swing — changing the effective length of the pendulum at twice its frequency. That is parametric resonance: an oscillation fed by periodically changing one of its own parameters, and fed most efficiently when the change runs at twice the oscillation’s frequency. The last curve to go found that in a system with no friction, orbits lie on smooth curves in phase space until the nonlinearity is strong enough to break them, at which point a chaotic sea opens between the surviving curves.

A weight on a spring, free to swing as well as bounce, does both of those things with nothing outside it. The bounce changes the spring’s length, and so the length of the pendulum it forms, at the bounce frequency. If that frequency is twice the swing’s, the bounce is a pump for the swing, and the energy in one passes to the other. And when there is enough energy for the exchange to become violent, the orderly passing breaks into chaos. The swinging spring, analysed by Aleksandr Vitt and Grigory Gorelik in 1933 as a mechanical model of a resonance in the vibrations of the carbon dioxide molecule, is the simplest system in which both can be watched with a weight, a spring and a ruler.

Two motions that should be independent #

A weight on a spring hanging from a fixed point has two obvious ways to move. It can bounce, the spring stretching and contracting along the vertical, at the spring frequency $\omega_s = \sqrt{k/m}$. Or it can swing, as a pendulum whose length is the stretched spring, at the pendulum frequency $\omega_p = \sqrt{g/\ell}$. For small motions the two are independent: a bounce does not tilt the spring, and a small swing does not stretch it to first order.

At second order they couple. A swinging weight pulls on the spring hardest at the bottom of each swing, where it is moving fastest, and least at the ends, so a swing stretches the spring twice in every cycle: a swing at $\omega_p$ drives the bounce at $2\omega_p$. And a bounce changes the pendulum’s length at $\omega_s$, which is a parametric drive for the swing, most effective when $\omega_s = 2\omega_p$. When the spring’s stiffness is chosen so that the bounce frequency is exactly twice the swing frequency, each motion drives the other at the frequency that feeds it.

Why twice, and not once #

The factor of two is the signature of parametric driving, and it is worth seeing why the obvious guess — that a bounce at the swing’s own frequency would feed it best — is wrong. A swing is fed when its length is shortened as the weight passes through the bottom, where the string’s tension is greatest and shortening it does the most work, and lengthened at the ends, where the tension is least and lengthening it costs little. The bottom comes twice per swing, once on each pass, so the length must go short-long-short-long at twice the swing frequency. A bounce at the swing’s own frequency would shorten the spring at the bottom of one pass and lengthen it at the bottom of the next, and gain nothing.

Stated in general, a parametrically driven oscillator is fed in bands of driving frequency centred on $2\omega_0/n$ for whole numbers $n$, the “tongues” of the Mathieu equation, and the widest and strongest is the first, at $2\omega_0$. Held up by a force that averages to nothing met the same equation from its other side, a pendulum whose pivot is shaken fast enough to stand it upside down — stability rather than growth, from a drive far above the tongues. The swinging spring sits in the first tongue, with the drive supplied not by a hand at the pivot but by the spring’s own bounce.

A bounce that turns into a swing #

The figure shows what happens. Pull the weight straight down by eight per cent of the spring’s length, tilt it by a hundredth of a radian or so — less than any hand could avoid — and let go. If the bounce is twice the swing frequency, the bounce does not stay a bounce. The tiny tilt grows: each bounce lengthens the pendulum as the weight passes the bottom of its small swing and shortens it at the ends, exactly the pumping the child on the swing does by standing and squatting, and the swing amplitude increases while the bounce dies. After about fifty units of time — eight swings — the bounce has gone completely and the weight is swinging sideways with all the energy. Then the process runs in reverse: the swing, stretching the spring twice per cycle, drives the bounce, which grows while the swing dies, and after a hundred units the weight is bouncing straight up and down again.

The demonstration needs almost nothing: a soft spring a few tens of centimetres long, a weight chosen so that its bounce, timed with a stopwatch, comes twice for each swing, and a little patience to adjust the weight until the exchange is strongest. The exchange repeats indefinitely, with nothing lost: the equations have no friction, and their energy, checked along every trajectory drawn here, is constant. Move the spring away from the resonance — make the bounce 2.4 times the swing rather than 2 — and almost nothing happens. The pumping is out of step with the swing, as a child standing at random moments on a swing gains nothing, and the swing never takes more than about one and a half per cent of the energy.

The path the bob draws #

The path of the weight records the exchange. It starts as a short vertical line, the bounce. As the swing grows, the line opens into a curved figure, wide at the top and narrow at the bottom: at the bottom of each swing the spring is stretched most, because the weight is moving fastest there, so the arcs dip. As the energy returns to the bounce, the figure narrows back to a line.

In three dimensions the spring does something more. The swing that grows out of a bounce picks some vertical plane, and from one cycle of exchange to the next the plane of the swing turns, so that a spring bouncing and swinging for a long time sweeps its swing round the vertical, like a Foucault pendulum on a much shorter timescale. The rate of that turning was worked out in the 1990s and found to depend on the angular momentum of the initial motion about the vertical in a way that is a mechanical analogue of a geometric phase. The two-dimensional motion drawn here has no such turning, since it starts with no angular momentum about the vertical.

Where regular exchange gives way to chaos #

The orderly exchange is a property of small amplitudes, where the coupling between bounce and swing is a weak correction to two independent oscillators. With more energy the coupling is no longer weak, and the question is what replaces the exchange.

Two things go wrong for the orderly exchange as the energy rises. The pendulum’s own period lengthens with its amplitude, as the period that depends on the swing found for any pendulum swung through a large angle, so a large swing falls out of the 2:1 ratio with the bounce that is supposed to pump it; and the spring’s stretch at the bottom of a large swing is no longer small, so the bounce and swing are no longer two oscillations with a weak coupling but one motion of a weight on an elastic string. The resonance that organised the small-amplitude motion is detuned by the motion itself, and the detuning depends on how the energy happens to be divided at each moment — which is the kind of feedback that turns an exchange into chaos.

A Poincaré section answers the question of what replaces it. Run the spring for a long time and, every time the spring passes its resting length while lengthening, record the tilt and the swing rate. A regular motion, confined to a smooth surface in the four-dimensional space of positions and velocities, leaves its dots on a closed curve. A chaotic one scatters them.

At low energy every orbit traces a curve. The pattern of curves has a cross at its centre: orbits that start almost entirely as bounce pass close to the point of no swing, and the arms of the cross are the exchange — swing growing out of nothing in one direction or the other. This is the regular resonance seen in the time series, now seen all at once for every starting condition.

At an energy of one — enough to swing the weight through a large angle and stretch the spring substantially — the picture is different. A few islands of regular motion survive, ringed by closed curves, but most of the section is filled with a scatter of dots that never settle onto any curve: a single orbit, followed long enough, wanders through the whole region. This is the chaotic sea that the last curve to go found opening in a kicked rotor as its kicks strengthened, appearing here in a weight on a spring as its energy rises.

Sensitivity to the start #

The practical meaning of the scatter is that two springs started almost identically do not stay together. Start two springs with tilts that differ by a hundred-millionth of a radian. At low energy the difference oscillates as the two exchanges drift slightly out of phase and grows only slowly; after two hundred units of time it is still about $10^{-7}$. At an energy of one the difference grows exponentially, by a factor of about ten every fifteen units of time, until after about a hundred and fifty units — some twenty-five swings — the two springs are doing entirely different things. No measurement of a real spring’s starting position is accurate to a hundred-millionth of a radian, so at that energy the spring’s motion beyond a couple of dozen swings cannot be predicted at all, although every step of it follows from Newton’s laws.

The rate of growth, a factor of ten in about fifteen units of time, is the spring’s largest Lyapunov exponent, the number the error that doubles on a schedule used to turn any uncertainty in a chaotic system’s state into a horizon beyond which prediction fails: each extra decimal place of starting precision buys only fifteen more units of time.

The wrong orbit beside a right one found the consolation that chaotic systems offer: a computed trajectory, wrong in detail, stays close to some true trajectory from a slightly different start. The swinging spring’s chaotic orbits have that property too, which is why their statistical behaviour can be computed even though their particular futures cannot.

The molecule the spring was built to explain #

Vitt and Gorelik’s paper was about a spring, but its motivation was a molecule. In carbon dioxide, the symmetric stretching vibration — both oxygen atoms moving in and out together — has very nearly twice the frequency of the bending vibration. Enrico Fermi showed in 1931 that this near-coincidence couples the two, so that the molecule’s infrared and Raman spectra show two lines where one would be expected, each a mixture of stretch and bend. That Fermi resonance is the quantum version of the spring’s exchange: two vibrations with frequencies in the ratio two to one, coupled at second order, sharing their character.

The same structure turns up wherever two oscillations have commensurate frequencies and a nonlinear coupling. The energy that refuses to be shared found a chain of nonlinear springs returning its energy to the mode it started in rather than spreading it across all of them, and the condition three modes never meet found that energy passes between waves only when their frequencies and wavenumbers satisfy a resonance condition. The swinging spring is the smallest version: two modes, one resonance, and a complete exchange.

A spring that models the weather #

The swinging spring has a second career, in meteorology. The atmosphere supports two very different kinds of motion: slow, large-scale flows that carry the weather, and fast gravity waves, ripples on the density layers of the air, that carry little of it. A forecast started from observed winds and pressures is contaminated by spurious fast waves, which swamp the slow evolution unless they are removed, and the removal — initialisation — has to keep the slow motion intact while killing the fast. Peter Lynch showed in 2002 that the swinging spring, at small amplitude, is a miniature of the same problem: the swing is the slow motion, the bounce the fast one, and the nonlinear coupling between them is the reason that simply setting the fast motion to zero at the start does not keep it at zero. Methods of balancing the spring’s two motions carry over to balancing a forecast’s.

What the model leaves out #

Friction. A real spring and a real pivot dissipate energy, and the exchange gradually weakens. It also biases it: the swing usually loses energy faster than the bounce, through air drag on a weight moving sideways at speed, so a real swinging spring left alone tends to end its life bouncing. With modest damping the exchange still completes several times before the motion dies, which is what makes the demonstration work on a bench.

The spring’s own mass and shape. A real spring is not massless, and a long coiled spring has modes of its own — waves travelling along the coil — that can join the exchange. The model treats the spring as an ideal massless element with a single stiffness.

Stiffness to bending. A real spring resists bending at its top, so it is not a perfect pivot, and its swing frequency is slightly higher than a simple pendulum’s. Tuning a real swinging spring to 2:1 is done by adjusting the mass until the exchange is seen to be strongest, rather than by calculation.

Three dimensions. The motion is confined here to one vertical plane. In three dimensions the plane of the swing precesses, as above, and at high energy the chaotic region is larger, since there are more ways for the motion to wander.

Still open: how the swing’s plane turns #

The swinging spring has been studied for ninety years and is still being studied, because it is simple enough to analyse completely at small amplitude and rich enough to show nearly every phenomenon of nonlinear oscillation. At small amplitude its motion can be written in closed form, using the averaged equations for the slowly varying amplitudes of the bounce and the swing, and those equations turn out to be integrable. The step-wise turning of the swing’s plane from one exchange to the next was explained in 2002 by Peter Lynch and Darryl Holm as a property of those averaged equations, with a turning angle per exchange set by a ratio of conserved quantities, and connected to the geometric phase of a system returning to its starting state by a closed path.

What happens between the small-amplitude regime, where the averaged equations hold, and the high-energy regime, where the section is chaotic, is less complete. The way the regular curves break up as the energy rises, which islands survive longest and at what energies the chaotic sea fills the section, have been mapped numerically for particular parameters but not explained in general, and the spring’s behaviour with damping and with a periodic drive at the top — the forms in which it is used to model coupled oscillations in engineering — is an active subject. The core phenomenon is settled. Two motions that look independent at small amplitude are coupled at the next order, and when their frequencies stand in the ratio of that coupling, the coupling wins: the bounce becomes a swing and the swing a bounce, regularly while the energy is small and chaotically when it is large.